It is apparent from the result in table 4.1 that the size of the spatial filter used to isolate the desired diffraction order must be considered as it can affect the accuracy of the desired wavefront. Too large a filter will allow through higher order harmonics and contaminate the beam whereas too small a filter will cut off the PSF and therefore not allow the full propagation of the beam. Furthermore it is seen that larger aberrations will require a larger spatial filter than smaller ones, and certain modes with more spatially extended PSFs will also. Regardless of the magnitude of the desired aberration however, the locations of the PSF for each harmonic will remain fixed as their positions are determined by the constant 60 waves tip and tilt applied with the intent of separating the orders. Figure 4.6 shows the PSF along with spatial filter for 1 wave astigmatism, 1 wave trefoil and the sum of both. It is apparent that the PSF for trefoil is significantly more spatially extended than for astigmatism alone, and even more so for the sum of the two. The spatial filter size therefore needs to be considered for each
CHAPTER 4. SIMULATED RECONFIGURABLE BINARY HOLOGRAMS 92
(a) (b) (c)
Figure 4.6: A comparison of the spatial filter and spatial extent of the PSF in the Fourier plane for an aberration of a) 1 wave astigmatism, b) 1 wave trefoil and c) 1 wave astigmatism + 1 wave trefoil.
aberration type being produced, especially for the case where multiple Zernike aberrations are being combined.
Figure 4.7 shows the growth of the recovered Z5 astigmatism component for a hologram of 5 waves RMS displayed on the SLM along with the total RMS of the produced wavefront. For a small spatial filter size, insufficient wavefront is passed through the system to produce the intended quantity of aberration until the spatial filter reaches approximately 100 pixels in size at which time the magnitude of the recovered aberration appears to plateau along with the RMS of the system. Further observation however shows that as the spatial filter continues to grow, more unwanted light is passed through the system resulting in growing artifacts in the recovered phase profile. These manifest themselves as jagged fringe edges in the observed interferogram as seen in figure 4.8. As one of the main purposes for using the SLM to create arbitrary phase profiles is use in nulling interferometry it is desirable that these artifacts be kept to a minimum as in a successful null test they would dominate the residual RMS of the wavefront.
It is not enough to merely consider the size of the aberration that the SLM is able to produce however. As mentioned above, one of the driving reasons for the consideration of such a device in wavefront shaping is the use of it in
Figure 4.7: Growth of recovered Z5 astigmatism component for a hologram of 5 waves RMS Z5 as a function of increasing spatial filter size. Until R=100 pixels the filter is too small to pass the wavefront fully through the system and above this value the recovered aberration quantity plateaus as errors caused by the contamination of higher orders passing through the filter are too small to dominate the RMS in the presence of the significantly larger astigmatism term. nulling interferometers. In such a case the created aberrations will cancel out existing aberrations to create an ideally flat interference pattern, however any small scale irregularities in wavefront production will show clearly and could potentially cause significant error by distorting this null pattern. Figure 4.9 shows the growth of the residual RMS error of the simulated wavefront with increasing spatial filter size where the SLM is displaying 5 waves astigmatism and then used to null out the same amount of aberration in the other arm of the interferometer. The high RMS seen at a filter size of under 100 pixels is due to insufficient PSF being passed through the system, however this residual drops sharply to roughly λ/100 once the PSF is entirely transmitted. Above this filter
CHAPTER 4. SIMULATED RECONFIGURABLE BINARY HOLOGRAMS 94 size however a gradual increase in the residual RMS of the system is seen as extra light makes it through the increasingly large pinhole. This light serves no purpose in the reconstruction of the desired wavefront and merely contam- inates it with higher order structure which during the null testing procedure dominates the fringe pattern. It should be noted that there is no flat bottom to figure 4.9 which tells us that there is a unique optimum spatial filter size for the specific aberration being considered, rather than a range of values which provide a minimum of higher order contamination. We expect that eventually the need for a large spatial filter for increasingly aspheric wavefronts will limit the possible production of highly aberrated profiles as we will begin to pass light from the zeroth order term. By increasing the carrier tilt we can further separate the distance between each diffracted order, however this then may give rise to insufficient sampling of the wavefront on the hologram display.
Figure 4.10 shows the null interference of 5 waves astigmatism with the wave- front produced from a hologram of the same aberration created with a spatial filter of 300 pixels in size, i.e. significantly larger than needed and giving a high residual RMS as seen in figure 4.9. It is clear from this that there is a large astigmatic artefact present in the image which dominates the phase profile. This contamination would mask small scale phase variations in a real experimental null test in the area of the artefact, an effect which can be minimised by re- ducing the spatial filter size. Notably it would appear this artefact can only be minimised through spatial filter variations , and not eliminated entirely.
Figure 4.8: Comparison of interferograms for 5 waves of Z5 astigmatism for two different spatial filter sizes. Left: Spatial filter radius=100 pixels, the first value for which the recovered wavefront approaches 5 waves. Right: Spatial filter R=380 which is beyond sufficient for full passage of the wavefront.
Figure 4.9: Growth of the residual error in recovered wavefront after a direct subtraction of 5 waves Z5 astigmatism for a varying spatial filter size.
CHAPTER 4. SIMULATED RECONFIGURABLE BINARY HOLOGRAMS 96
Figure 4.10: Left: Interferogram of a null test between a perfect 5 waves of Zernike astigmatism and the wavefront produced by displaying 5 waves on the SLM and using an F=300 pixel spatial filter. Right: Unwrapped phase profile of interferogram on left showing a residual RMS of 0.017 waves. Scale on key in fractions of a wave.